M. Golubitsky and I.N. Stewart
Hopf bifurcation with dihedral group symmetry: coupled nonlinear oscillators
In: Multiparameter Bifurcation Theory. (M. Golubitsky and J. Guckenheimer, eds.) Contemporary Mathematics 56 AMS, 1986, 131-173.
We apply the theory of Hopf bifurcation with symmetry developed in Golubitsky and Stewart (1985) to systems of ODEs having the symmetries of a regular polygon, that is, whose symmetry group is dihedral. We consider the existence and stability of symmetry-breaking branches of periodic solutions. In particular we apply these results to a general system of n nonlinear oscillators, coupled symmetrically in a ring, and describe the generic oscillation patterns. We find, for example, that the symmetry can force some oscillators to have twice the frequency of others. The case of four oscillators has exceptional features.